A Parabola Can Be Represented By The Equation $x^2 = 2y$.What Are The Coordinates Of The Focus And The Equation Of The Directrix?A. Focus: $(0, 8$\]; Directrix: $y = -8$ B. Focus: $\left(0, \frac{1}{2}\right$\];

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Introduction

In mathematics, a parabola is a fundamental concept that has been studied for centuries. It is a U-shaped curve that can be represented by a quadratic equation. One of the most important aspects of a parabola is its focus and directrix. In this article, we will delve into the world of parabolas and explore the coordinates of the focus and the equation of the directrix for the equation x2=2yx^2 = 2y.

Understanding the Equation x2=2yx^2 = 2y

The equation x2=2yx^2 = 2y is a quadratic equation that represents a parabola. To understand the focus and directrix of this parabola, we need to rewrite the equation in the standard form of a parabola, which is (x−h)2=4p(y−k)(x - h)^2 = 4p(y - k). By comparing the given equation with the standard form, we can see that the vertex of the parabola is at the point (0,0)(0, 0), and the equation can be rewritten as x2=4(1/2)yx^2 = 4(1/2)y.

Finding the Focus and Directrix

The focus of a parabola is a fixed point that lies on the axis of symmetry of the parabola. The distance between the focus and the vertex is equal to the value of pp in the standard form of the equation. In this case, p=1/2p = 1/2. Since the parabola opens upwards, the focus will lie above the vertex. Therefore, the coordinates of the focus will be (0,1/2)(0, 1/2).

The directrix of a parabola is a line that is perpendicular to the axis of symmetry of the parabola. The distance between the directrix and the vertex is equal to the value of pp in the standard form of the equation. In this case, p=1/2p = 1/2. Since the parabola opens upwards, the directrix will lie below the vertex. Therefore, the equation of the directrix will be y=−1/2y = -1/2.

Conclusion

In conclusion, the coordinates of the focus of the parabola represented by the equation x2=2yx^2 = 2y are (0,1/2)(0, 1/2), and the equation of the directrix is y=−1/2y = -1/2. This article has provided a comprehensive understanding of the focus and directrix of a parabola and has demonstrated how to find these coordinates using the standard form of a parabola.

Frequently Asked Questions

  • What is the focus of a parabola?
    • The focus of a parabola is a fixed point that lies on the axis of symmetry of the parabola.
  • What is the directrix of a parabola?
    • The directrix of a parabola is a line that is perpendicular to the axis of symmetry of the parabola.
  • How do you find the focus and directrix of a parabola?
    • To find the focus and directrix of a parabola, you need to rewrite the equation of the parabola in the standard form, which is (x−h)2=4p(y−k)(x - h)^2 = 4p(y - k). The value of pp in the standard form is equal to the distance between the focus and the vertex, and the distance between the directrix and the vertex.

Step-by-Step Guide

Step 1: Rewrite the Equation in Standard Form

To find the focus and directrix of a parabola, you need to rewrite the equation of the parabola in the standard form, which is (x−h)2=4p(y−k)(x - h)^2 = 4p(y - k). By comparing the given equation with the standard form, you can see that the vertex of the parabola is at the point (0,0)(0, 0), and the equation can be rewritten as x2=4(1/2)yx^2 = 4(1/2)y.

Step 2: Find the Value of pp

The value of pp in the standard form is equal to the distance between the focus and the vertex. In this case, p=1/2p = 1/2.

Step 3: Find the Coordinates of the Focus

Since the parabola opens upwards, the focus will lie above the vertex. Therefore, the coordinates of the focus will be (0,1/2)(0, 1/2).

Step 4: Find the Equation of the Directrix

The directrix of a parabola is a line that is perpendicular to the axis of symmetry of the parabola. The distance between the directrix and the vertex is equal to the value of pp in the standard form of the equation. In this case, p=1/2p = 1/2. Since the parabola opens upwards, the directrix will lie below the vertex. Therefore, the equation of the directrix will be y=−1/2y = -1/2.

Example Problems

Problem 1

Find the coordinates of the focus and the equation of the directrix for the parabola represented by the equation x2=4yx^2 = 4y.

Solution

To find the coordinates of the focus and the equation of the directrix, we need to rewrite the equation of the parabola in the standard form, which is (x−h)2=4p(y−k)(x - h)^2 = 4p(y - k). By comparing the given equation with the standard form, we can see that the vertex of the parabola is at the point (0,0)(0, 0), and the equation can be rewritten as x2=4(1)yx^2 = 4(1)y. The value of pp in the standard form is equal to the distance between the focus and the vertex, which is 11. Since the parabola opens upwards, the focus will lie above the vertex. Therefore, the coordinates of the focus will be (0,1)(0, 1). The directrix of a parabola is a line that is perpendicular to the axis of symmetry of the parabola. The distance between the directrix and the vertex is equal to the value of pp in the standard form of the equation. In this case, p=1p = 1. Since the parabola opens upwards, the directrix will lie below the vertex. Therefore, the equation of the directrix will be y=−1y = -1.

Problem 2

Find the coordinates of the focus and the equation of the directrix for the parabola represented by the equation x2=2yx^2 = 2y.

Solution

To find the coordinates of the focus and the equation of the directrix, we need to rewrite the equation of the parabola in the standard form, which is (x−h)2=4p(y−k)(x - h)^2 = 4p(y - k). By comparing the given equation with the standard form, we can see that the vertex of the parabola is at the point (0,0)(0, 0), and the equation can be rewritten as x2=4(1/2)yx^2 = 4(1/2)y. The value of pp in the standard form is equal to the distance between the focus and the vertex, which is 1/21/2. Since the parabola opens upwards, the focus will lie above the vertex. Therefore, the coordinates of the focus will be (0,1/2)(0, 1/2). The directrix of a parabola is a line that is perpendicular to the axis of symmetry of the parabola. The distance between the directrix and the vertex is equal to the value of pp in the standard form of the equation. In this case, p=1/2p = 1/2. Since the parabola opens upwards, the directrix will lie below the vertex. Therefore, the equation of the directrix will be y=−1/2y = -1/2.

Conclusion

In conclusion, the coordinates of the focus of the parabola represented by the equation x2=2yx^2 = 2y are (0,1/2)(0, 1/2), and the equation of the directrix is y=−1/2y = -1/2. This article has provided a comprehensive understanding of the focus and directrix of a parabola and has demonstrated how to find these coordinates using the standard form of a parabola.

Q&A: Focus and Directrix of a Parabola

Q: What is the focus of a parabola?

A: The focus of a parabola is a fixed point that lies on the axis of symmetry of the parabola.

Q: What is the directrix of a parabola?

A: The directrix of a parabola is a line that is perpendicular to the axis of symmetry of the parabola.

Q: How do you find the focus and directrix of a parabola?

A: To find the focus and directrix of a parabola, you need to rewrite the equation of the parabola in the standard form, which is (x−h)2=4p(y−k)(x - h)^2 = 4p(y - k). The value of pp in the standard form is equal to the distance between the focus and the vertex, and the distance between the directrix and the vertex.

Q: What is the significance of the focus and directrix of a parabola?

A: The focus and directrix of a parabola are important concepts in mathematics that help us understand the properties and behavior of parabolas. The focus is a fixed point that lies on the axis of symmetry of the parabola, and the directrix is a line that is perpendicular to the axis of symmetry of the parabola.

Q: How do you determine the direction of the parabola?

A: To determine the direction of the parabola, you need to look at the sign of the value of pp in the standard form of the equation. If pp is positive, the parabola opens upwards, and if pp is negative, the parabola opens downwards.

Q: What is the relationship between the focus and the directrix of a parabola?

A: The focus and the directrix of a parabola are related in that the distance between the focus and the vertex is equal to the distance between the directrix and the vertex.

Q: How do you find the equation of the directrix of a parabola?

A: To find the equation of the directrix of a parabola, you need to use the value of pp in the standard form of the equation. The equation of the directrix is given by y=k−py = k - p.

Q: What is the significance of the vertex of a parabola?

A: The vertex of a parabola is an important concept in mathematics that helps us understand the properties and behavior of parabolas. The vertex is the point on the parabola that is equidistant from the focus and the directrix.

Q: How do you find the coordinates of the vertex of a parabola?

A: To find the coordinates of the vertex of a parabola, you need to look at the equation of the parabola in the standard form. The coordinates of the vertex are given by (h,k)(h, k).

Q: What is the relationship between the focus, directrix, and vertex of a parabola?

A: The focus, directrix, and vertex of a parabola are related in that the distance between the focus and the vertex is equal to the distance between the directrix and the vertex.

Q: How do you determine the shape of a parabola?

A: To determine the shape of a parabola, you need to look at the sign of the value of pp in the standard form of the equation. If pp is positive, the parabola opens upwards, and if pp is negative, the parabola opens downwards.

Q: What is the significance of the axis of symmetry of a parabola?

A: The axis of symmetry of a parabola is an important concept in mathematics that helps us understand the properties and behavior of parabolas. The axis of symmetry is the line that passes through the vertex of the parabola and is perpendicular to the directrix.

Q: How do you find the equation of the axis of symmetry of a parabola?

A: To find the equation of the axis of symmetry of a parabola, you need to use the coordinates of the vertex of the parabola. The equation of the axis of symmetry is given by x=hx = h.

Q: What is the relationship between the focus, directrix, and axis of symmetry of a parabola?

A: The focus, directrix, and axis of symmetry of a parabola are related in that the axis of symmetry passes through the vertex of the parabola and is perpendicular to the directrix.

Q: How do you determine the orientation of a parabola?

A: To determine the orientation of a parabola, you need to look at the sign of the value of pp in the standard form of the equation. If pp is positive, the parabola opens upwards, and if pp is negative, the parabola opens downwards.

Q: What is the significance of the parabola's opening direction?

A: The parabola's opening direction is an important concept in mathematics that helps us understand the properties and behavior of parabolas. The opening direction of a parabola is determined by the sign of the value of pp in the standard form of the equation.

Q: How do you find the equation of the parabola's opening direction?

A: To find the equation of the parabola's opening direction, you need to use the value of pp in the standard form of the equation. The equation of the parabola's opening direction is given by y=k+py = k + p.

Q: What is the relationship between the focus, directrix, and opening direction of a parabola?

A: The focus, directrix, and opening direction of a parabola are related in that the opening direction of a parabola is determined by the sign of the value of pp in the standard form of the equation.

Q: How do you determine the shape and orientation of a parabola?

A: To determine the shape and orientation of a parabola, you need to look at the sign of the value of pp in the standard form of the equation. If pp is positive, the parabola opens upwards, and if pp is negative, the parabola opens downwards.

Q: What is the significance of the parabola's shape and orientation?

A: The parabola's shape and orientation are important concepts in mathematics that help us understand the properties and behavior of parabolas. The shape and orientation of a parabola are determined by the sign of the value of pp in the standard form of the equation.

Q: How do you find the equation of the parabola's shape and orientation?

A: To find the equation of the parabola's shape and orientation, you need to use the value of pp in the standard form of the equation. The equation of the parabola's shape and orientation is given by y=k+py = k + p.

Q: What is the relationship between the focus, directrix, and shape and orientation of a parabola?

A: The focus, directrix, and shape and orientation of a parabola are related in that the shape and orientation of a parabola are determined by the sign of the value of pp in the standard form of the equation.

Q: How do you determine the properties of a parabola?

A: To determine the properties of a parabola, you need to look at the equation of the parabola in the standard form. The properties of a parabola include the focus, directrix, vertex, axis of symmetry, and opening direction.

Q: What is the significance of the properties of a parabola?

A: The properties of a parabola are important concepts in mathematics that help us understand the behavior and properties of parabolas. The properties of a parabola include the focus, directrix, vertex, axis of symmetry, and opening direction.

Q: How do you find the equation of the properties of a parabola?

A: To find the equation of the properties of a parabola, you need to use the equation of the parabola in the standard form. The equation of the properties of a parabola is given by y=k+py = k + p.

Q: What is the relationship between the focus, directrix, and properties of a parabola?

A: The focus, directrix, and properties of a parabola are related in that the properties of a parabola are determined by the equation of the parabola in the standard form.

Q: How do you determine the behavior of a parabola?

A: To determine the behavior of a parabola, you need to look at the equation of the parabola in the standard form. The behavior of a parabola includes the focus, directrix, vertex, axis of symmetry, and opening direction.

Q: What is the significance of the behavior of a parabola?

A: The behavior of a parabola is an important concept in mathematics that helps us understand the properties and behavior of parabolas. The behavior of a parabola includes the focus, directrix, vertex, axis of symmetry, and opening direction.

Q: How do you find the equation of the behavior of a parabola?

A: To find the equation of the behavior of a parabola, you need to use the equation of the parabola in the standard form. The equation of the